Ordered structures of constructing operators for generalized Riesz systems (Q1728983)

Summary: A sequence \(\{\varphi_n \}\) in a Hilbert space \(\mathcal H\) with inner product \(\langle \cdot, \cdot \rangle\) is called a generalized Riesz system if there exist an ONB \(\boldsymbol{e} = \{e_n \}\) in \(\mathcal H\) and a densely defined closed operator \(T\) in \(\mathcal H\) with densely defined inverse such that \(\{e_n \} \subset D(T) \cap D((T^{- 1})^\ast)\) and \(T e_n = \varphi_n\), \(n = 0,1, \dots\), and \((\boldsymbol{e}, T)\) is called a constructing pair for \(\{\varphi_n \}\) and \(T\) is called a constructing operator for \(\{\varphi_n \}\). The main purpose of this paper is to investigate under what conditions the ordered set \(C_\varphi\) of all constructing operators for a generalized Riesz system \(\{\varphi_n \}\) has maximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications.